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Inverse Problems and Imaging (IPI)
 

Correlation priors

Pages: 167 - 184, Volume 5, Issue 1, February 2011      doi:10.3934/ipi.2011.5.167

 
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Lassi Roininen - University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland (email)
Markku S. Lehtinen - University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland (email)
Sari Lasanen - University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä, Finland (email)
Mikko Orispää - Sodankylä Geophysical Observatory, University of Oulu, Tähteläntie 62, FIN-99600 Sodankylä, Finland (email)
Markku Markkanen - Eigenor Corporation, Lompolontie 1, FI-99600 Sodankylä, Finland (email)

Abstract: We propose a new class of Gaussian priors, correlation priors. In contrast to some well-known smoothness priors, they have stationary covariances. The correlation priors are given in a parametric form with two parameters: correlation power and correlation length. The first parameter is connected with our prior information on the variance of the unknown. The second parameter is our prior belief on how fast the correlation of the unknown approaches zero. Roughly speaking, the correlation length is the distance beyond which two points of the unknown may be considered independent.
   The prior distribution is constructed to be essentially independent of the discretization so that the a posteriori distribution will be essentially independent of the discretization grid. The covariance of a discrete correlation prior may be formed by combining the Fisher information of a discrete white noise and different-order difference priors. This is interpreted as a combination of virtual measurements of the unknown. Closed-form expressions for the continuous limits are calculated. Also, boundary correction terms for correlation priors on finite intervals are given.
   A numerical example, deconvolution with a Gaussian kernel and a correlation prior, is computed.

Keywords:  Statistical inversion, a priori distributions, discretization-invariance.
Mathematics Subject Classification:  Primary: 65Q10, 60G10; Secondary: 42A38.

Received: April 2009;      Revised: October 2010;      Available Online: February 2011.

 References